Non-extremal geometries and holographic phase transitions

# Non-extremal geometries and holographic phase transitions

## Abstract:

Using the low energy limit of type IIB superstring theory, we obtain the non-extremal limit of deformed conifold geometry which is dual to the IR limit of large N thermal QCD. At low temperatures, the extremal geometry without black hole is favored while at high temperatures, the field theory is described by non-extremal black hole geometry. We compute the ten dimensional on shell action for extremal and non-extremal geometries and demonstrate that at a critical temperature there is a first order confinement to deconfinement phase transition. We compute as a function of ’tHooft coupling and study the thermodynamics of the dual gauge theory by evaluating the free energy and entropy of the ten dimensional geometry. We find agreement with the conformal limit while thermodynamics of non-conformal strongly coupled gauge theories is explored using the black hole geometries in non-AdS space.

## 1 Introduction

At extreme temperatures nuclear matter is best described by a weakly interacting gas of gluons and quarks due to asymptotic freedom [1, 2]. The weak coupling allows one to use perturbative techniques to study the thermodynamics of the system and at zeroth order in perturbation theory, it is best described as a gas of free particles. But the same asymptotic freedom implies that the perturbative analysis must break down with decreasing temperature. As temperature is lowered, the couplings get stronger and color degrees of freedom are confined. Thus nuclear matter undergoes a phase transition- from low temperature confined phase of color neutral constituents to high temperature deconfined phase of quarks and gluons. However to analyze the non-perturbative confinement mechanism, one has to study the theory on the lattice [3, 4, 5] or resort to effective field theories - both of which are successful along with their inherent limitations1.

On the other hand ’tHooft showed that when number of colors goes to infinity, only planar diagrams contribute to the amplitude [10, 11]. Thus in the large limit, the theory drastically simplifies and one may hope that the thermodynamics could be analytically tractable. Furthermore, the structure of the planar diagrams suggest that four dimensional gauge theory maybe interpreted as a string theory [12]. This connection becomes clear with the realization that gauge theories naturally arise from excitations of strings ending on branes [35], while on the other hand gravitons arise from low energy excitations of closed strings. By studying the interactions between open and closed strings, the correspondence between gauge theory and gravity can be realized. The best studied example is the AdS/CFT correspondence proposed by Maldacena [36] which essentially maps maximally supersymmetric conformal gauge theory to geometry. In the large limit, the gauge theory has large ’tHooft coupling and geometry with small curvature has a classical description. Again the appearance of large has allowed an exact analytic description of strongly coupled quantum gauge theory in terms of weakly coupled dual classical gravity.

In [37] an exact proposal was made to compute correlation functions of four dimensional conformal gauge theory using partition function of supergravity on AdS space. To study the thermodynamic properties of the gauge theory, one studies the thermodynamics of the dual geometry and for the case of CFTs, this amounts to the study of black holes in AdS space. The AdS/CFT correspondence has been quite successful in this regard- the scaling of thermodynamic state functions (free energy, pressure, entropy) of CFTs with respect to temperature matches with that of black holes [38, 39]. However there are no phase transitions in conformal field theories and the strong coupling regime of QCD matter is far from conformal. Thus the study of confinement to deconfinement phase transition in QCD requires careful extension of the gauge/gravity correspondence to incorporate the conformal anomalies.

In principal this can be done by placing D branes in ten dimensional geometries and then studying the warped geometry sourced by the fluxes and scalar fields arising from the branes. Since QCD is non supersymmetric and non-conformal, the first objective is to find gauge theories with RG flows arising from D brane configurations with minimal SUSY. There have been a lot of progress in that direction: In [41, 42] RG flows that connected conformal fixed points at IR and UV was incorporated, and [43] connected the UV conformal fixed point to a confining theory. But the model with QCD like logarithmic running of the coupling and minimal supersymmetry is the Klebanov-Strassler (KS) model [44] (with an extension by Ouyang [45] to incorporate fundamental matters). Although at the highest energies the gauge theory is best described in terms of bifundamental fields with effective degrees of freedom diverging, at the lowest energies the gauge theory resembles SUSY QCD. Again in the limit when effective brane charge is large, the gauge theory with large ’tHooft coupling has an equivalent description in terms of warped deformed cone.

The thermodynamics of this non-conformal gauge theory is encoded in the dual geometry. A great deal of effort has been given in computing the black hole geometry dual to non-conformal thermal gauge theories. For example in [46, 47, 48, 49] the cascading picture of the original KS model was extended to incorporate black-hole without any fundamental matter, while fundamental matter was accounted for in [50]. Most of the attempts are based on obtaining an effective lower dimensional action from KK reducing ten dimensional supergravity action. Dimensional reduction of a generic ten dimensional action is a formidable challenge specially when there are non-trivial fluxes and scalar fields and it is highly non-trivial to obtain a consistent truncation. Furthermore, it is not clear how RG flow of the dual gauge theory can be obtained since the fields in the effective action are not the dilaton or the flux in the original ten dimensional action.

On the other hand, partition function of the geometry and thus the thermodynamics of the dual gauge theory can be directly obtained by computing the on shell gravity action with appropriate boundary terms [51]. In a series of papers [52]-[58], we proposed the ten dimensional deformed resolved conifold black hole geometry as the dual to UV complete gauge theory which resembles large thermal QCD. Working with the ten dimensional geometry, we avoid the difficulty of KK reduction while extracting the exact RG flow of the gauge theory. The thermal gauge theory studied in [52]-[58] has a rich phase structure and as temperature is altered, we expect phase transitions. Our goal is to study the phase transitions at strong coupling, but to do so, we must first understand how geometries can describe different phases. In this work, we make progress in that direction and obtain the most general ten dimensional geometry (with or without a black hole) that arises from low energy limit of type IIB superstring theory. Directly identifying the gauge theory partition function with that of the ten dimensional geometry, we show how phase transition is realized. For any given temperature of the dual gauge theory, there are two geometries extremal (without black hole) and non-extremal (with black hole) but the geometry with lower on shell action is preferred. At a critical temperature , both geometries are equally likely and we have a phase transition.

Our ten dimesional solution is analytic and the corrections to the metric due to the black hole can be exactly written as a Taylor series in and where is the Schwarzchild horizon and are number of five branes and effective number of three branes at some high energy. This series expansion allows us to write down exact expression for the on shell gravity action with or without the black hole and then obtain the critical horizon. However, in our analysis of the non-extremal geometry, we have considered constant axio-dilaton field. There are no D7 branes, no fundamental matter in the dual thermal gauge theory and no Baryochemical potential. On the other hand, it is straight forward to incorporate chemical potential in extremal ten dimensional geometry by considering gauge fluxes on holomorphically embedded D7 branes. Note that D7 brane embedding in black hole geometry is highly non-trivial in the presence of non-trivial three form fluxes2. What we have been able to obtain is the black hole geometry with only non-trivial five form and three form but constant dilaton and already in this simplified scenario, we find a first order phase transition. Observe that the warped deformed cone of KS model is dual to , pure gauge theory which has no fundamental matter but it confines, exhibits a mass gap and has gluino condensates [59]. Thus the confinement/deconfinement phase transition we obtain should mimic the phase transition between confined glue-ball and deconfined gluons of QCD without flavor. In our upcoming work [60], we will incorporate the effect of running dilaton field and other localized sources on the non-extremal geometry which will also give the dual geometric description of UV complete thermal gauge theory.

In section 2, we briefly discuss the supergravity equations and outline the procedure of obtaining thermodynamic state functions of the gauge theory. Summarizing the extremal geometry and its dual confined gauge theory in section 2.1, we propose the non-extremal limit of ten dimensional geometry in section 2.2. Using the supergravity solutions for extremal and non-extremal geometries, in section 3, we study transitions between the geometries which essentially describes phase transitions in the dual gauge theory. When the boundary of the geometry scales with the ’tHooft coupling , we find the critical temperature as a function of . Finally in section 4, we discuss the structure of the gauge theory and propose a possible brane configuration that gives rise to the supergravity solutions. Since our ultimate goal is to learn about thermal phase transitions in nuclear matter, we describe the connections between the gauge theory and large thermal QCD.

## 2 Gravity Action and Gauge Theory

We start with the type IIB supergravity action including local sources in ten dimensions [61][62]:

 Stotal = SSUGRA+Sloc=12κ210∫d10x√G(R+∂M˜τ∂M¯˜τ2|Im˜τ|2−|˜F5|24⋅5!−G3⋅¯G312Im˜τ) (1) + ∫Σ8C4∧R(2)∧R(2)+18iκ210∫C4∧G3∧¯G3Im˜τ+Sloc

in Einstein frame, where is the axio-dilaton with being the axion and the dilaton field and is the five-form flux sourced by the D3 or fractional D3 branes. Here with the RR three form flux and the NS-NS three form flux with being the NS-NS two form while is the RR two form. We also have the four-form potential, with being the metric, the curvature two-form, and is the action for localized sources in the system (i.e. D branes).

The above action (1) is the most general supergravity action obtained from type IIB superstring action with fluxes and localized sources. Minimizing the action leads to the following Einstein equations

 Rμν = −gμν⎡⎣G3⋅¯G348Im˜τ+˜F258⋅5!⎤⎦+˜Fμabcd˜Fabcdν4⋅4!+κ210(Tlocμν−18gμνTloc) Rmn = −gmn⎡⎣G3⋅¯G348Im˜τ+˜F258⋅5!⎤⎦+˜Fmabcd˜Fabcdn4⋅4!+Gbcm¯Gnbc4Im˜τ+∂m˜τ∂n˜τ2|Im˜τ|2 (2) + κ210(Tlocmn−18gmnTloc)

where ; and we have assumed that the fluxes and axio-dilaton only depends on coordinates .

The equation of motion for can be expressed in terms of a seven-form in the following way,

 dΛ7−iIm˜τd˜τ∧ReΛ7=0 (3)

where typically quantifies the deviations from the imaginary self dual (ISD) behavior. Minimizing the action (1) also gives the Bianchi identity for the five-form flux [62]

 d˜F5=−G3∧¯G32iIm˜τ+2κ210T3ρloc3 (4)

where is the D3 charge density from the localized sources and is the brane tension. In the presence of D3 branes, is a delta function peaked at the location of the branes while D7 or fractional D5 branes may also contribute to D3 charge.

To keep lorentz invariance along the space-time direction we assume the self-dual five-form has the form

 ~F5=(1+∗)dα∧dt∧dx∧dy∧dz. (5)

where is a scalar field, function of the internal coordinates . We take a general metric ansatz:

 ds2 = −e2A+2Bdt2+e2A(dx2+dy2+dz2)+e−2A−2B~gmndxmdxn ~gmndxmdxn = (a(r)dr+k(r)ds2M5) (6)

where characterizes the existence of a black hole. Using this metric ansatz we can express as

 (7)

The above choice of leads us to three different classes of solutions from the EOM (3). These three classes can be tabulated in the following way:

If in (7) and then must be imaginary self dual (ISD). When then this is the same as GKP solution [62], and in this case is not restricted and

 ~Rmn=∂m˜τ∂n¯˜τ2∣Im˜τ∣2−3˜▽m∂nB−32∂mB∂nB (8)

where is the Ricci tensor for cone metric .

If then we can take but keep and . This means is a constant3.

If then we can again take but now and such that (3) is satisfied. This means both axion and the dilaton could run in this scenario.

The first possibility is studied in the appendix A. We are more interested in the last two options because it is well established that AdS black hole solution corresponds to conformal gauge theory, and non-AdS black hole solutions should reduce to AdS black hole solutions in certain limit. It is obvious that in the first case at it does not recover AdS black hole solution while the last two cases obviously do.

Suppose now we find a solution to the above set of Einstein equations (2) along with the flux equations (3), (4). The resulting geometry with the metric (2) has the topology of and in particular for , can be a five dimensional AdS space with or without black holes which has throat radius while can be a compact five sphere with radius . For non-vanishing three form flux , will be a non-AdS space and the size of given by will diverge for large . However for fixed radial coordinate , the geometry has topology where is four dimensional Minkowski space and has fixed radius. This way at fixed radial location (which we interpret as the boundary), we can obtain a four dimensional manifold by integrating over the compact space . The gauge theory is defined on Minkowski space while the manifold is the dual geometry for the gauge theory.

According to AdS/CFT correspondence, the partition function of string theory on should coincide with the partition function of super- Yang-Mills theory. For generalized gauge/gravity correspondence, we can identify the gauge theory partition function with that of ten dimensional dual geometry,

 Zgauge = e−F/T=Zgravity≃e−Srengravity Srengravity = Stotal+SGH+Scounter (9)

where are free energy and temperature of the gauge theory, is given by (1), is the Gibbons-Hawking boundary term [51] and is the counter term necessary to renormalize the action. The gravity action is evaluated on and then wick rotated to obtain the Euclidean on shell value. The is because we have ignored all the corrections and loop corrections.

Using (2), we can in principle obtain all the thermodynamic quantities for the four dimensional gauge theory by considering the ten dimensional action on shell. For instance, free energy and internal energy of the gauge theory are given by

 F = −T log(Zgauge)=−T log(Zgravity)=T Srengravity E = T2∂∂T(logZgauge)=∂Srengravity∂β (10)

where is the temperature. Knowing the free energy one gets the pressure and entropy

 p = −(∂F∂V3)T s = −(∂F∂T)V3 (11)

where is the volume of three dimensional flat space.

Our primary concern for this paper is to obtain the ten dimensional geometries that can arise from the action (1). However, the same action can give rise to more than one manifolds. In fact, just like the case for AdS space discussed by Hawking and Page [63] and elaborated by Witten [64], there are two manifolds and that minimizes the action (1). The manifold which has lower value for the on shell action for a given temperature of the dual gauge theory will be preferred. Since and are distinct geometries, the thermodynamics of the gauge theory will be different at different temperatures, depending on which geometry is preferred. This means and will correspond to different phases of the gauge theory and we will now analyze the manifolds in some details bellow.

### 2.1 Extremal geometry and confinement

The metric of the extremal geometry without any black hole, i.e. , is given by [44] (with Minkowski signature)

 ds2 = e2A[−dt2+dx2+dy2+dz2]+e−2A¯gmndxmdxn (12)

where and is the metric of the deformed cone

 +sinh2(ρ2)[(g1)2+(g2)2]] (13)

where is a constant, are one forms given by

 g1=e1−e3√2,    g2=e2−e4√2 g3=e1+e3√2,    g4=e2+e4√2,   g5=e5 e1≡−sinθ1dϕ1,    e2≡dθ1 e3≡cosψsinθ2dϕ2−sinψdθ2, e4≡sinψsinθ2dϕ2+cosψdθ2, e5≡dψ+cosθ1dϕ1+cosθ2dϕ2 (14)

and

 K(ρ)=(sinh(2ρ)−2ρ)1/321/3sinhρ. (15)

The three form fluxes on the deformed cone is Imaginary Self Dual (ISD) to preserve the supersymmetry while the five form fluxes is self dual

 ∗6G3=iG3, ˜F5=(1+∗)dh−1∧dt∧dx∧dy∧dz=dC4+B2∧F3. (16)

Note that the metric (2.1) is Ricci flat, that is the Ricci tensor for the metric denoted by (but Ricci tensor for the metric denoted by ), while minimizing the action (1) with non-zero and localized sources will give rise to an internal metric which is not Ricci flat in general. In particular, the Ricci tensor for the warped metric is

 Rμν = ημν14h(ρ)¯▽2logh(ρ) Rmn = ¯Rmn−¯gmn4~▽2logh(ρ)−12∂mlogh(ρ)∂nlogh(ρ) (17)

where

 ¯▽2=¯gmn∂m∂n+∂m¯gmn∂n+12¯gmn¯gpq∂n¯gpq∂m (18)

Now using (2.1) in (2) with the metric given by (12), one readily gets that . However, F-theory [66, 67] dictates that the running of the axio-dilation field is where is the number of seven branes and we will only consider terms which are of ignoring all higher order terms. If we also ignore the localized sources, then indeed we obtain [57, 62].

Even though remains the same the warp factor is different in different regions. In the IR region we assume the D7 branes are far away and thus the axion-dilaton is constant. The warp factor, and in this region are given in [44] and calculated to the first order. The warp factor can be written in the following form

 h(ρ)=ciρi (19)

where the coefficient can be treated as constants. When doing so it is assumed that

 ∗dh−1∧dt∧dx1∧dx2∧dx3=B2∧F3 (20)

This is correct if the number of D3 branes is multiple of that of the D5 branes. In the rest of the paper we will restrict to this special case4.

The warped deformed conifold with given by (19) for small as proposed in [44] in fact removes the IR singularity of the warped conifold [65] and gives rise to linear confinement in the dual gauge theory. Unlike the regular cone, the deformed cone has a blown up at the tip of the cone and this finite size of the removes the IR divergence of the fluxes [44]. Observe that near , the metric in (2.2) reduces to that of

 ¯gmndxmdxn∼A4/3(2/3)1/3[12(g5)2+(g3)2+(g4)2] (22)

which implies that the radius of at the tip of the cone is of . The finite size of at gives finite value for the three form flux strength at the tip of the cone. This way, the IR singularity of the fluxes are removed, which consequently removes the IR singularity of the warp factor.

On the other hand, this constant modifies the embedding equation for the cone and is related to the expectation values of gauge invariant operators5 in the dual gauge theory [44]. Hence a non-zero results in non-zero expectation value and corresponds to spontaneous breaking of group down to where is the number of fractional three branes placed at the tip of the cone. Note that in the absence of D7 branes, this spontaneous broken symmetry is not identical to the chiral symmetry of QCD as there are no fundamental flavor. But even in the presence of seven branes, the deformed cone metric (12) with warp factor (19) will be valid by considering the D7 branes as probes. The probe seven branes will give rise to fundamental matter and depending on their embedding, can lead to breaking of flavor chiral symmetry just like in QCD [68].

Coming back to the metric (2.1), observe that with a change of coordinates

 r3=A2eρ (23)

for large , the metric becomes

 ¯gmndxmdxn∼dr2+r2(19(g5)2+164∑i=1(gi)2) (24)

which is the metric of regular cone with base . Thus only for small radial coordinate , the internal metric is a deformed cone while at large , we really have a regular cone with topology of .

However for large , we can no longer ignore the running of the field as we will be near the seven branes. As we have a regular cone for large , we can use Ouyang’s holomorphic embedding of seven branes [45] to determine the running of the field along with the modified flux (which is again ISD) and . Then for , using change of coordinates (23), we obtain the warp factor

 (25) + 81g2sM2Nf32πlog(rrmin)log(sinθ12sinθ22))

where is the number of D7 branes which is present in the action (1) and source the field while is the minimum radial distance reached by the seven brane. The exact brane configuration that could give rise to such a warp factor in the dual geometry will be discussed in section 4. Also are constants which will be determined by matching the above solution for large with the solution (19) valid at small . In fact is proportional to the D3 brane charge at . To see this we notice that on the gravity action, we do not have any branes - only fluxes. However, due to the duality, the units of the fluxes on the gravity side should equal to the number of branes on the gauge theory side. This way the units of five form fluxes can be identified as the effective charge:

 Neff = 12κ210T3∫T1,1~F5=−1216κ210T3∫r5dhdrg1∧g2∧g3∧g4∧g5 (26)

where is the three brane tension and the Gaussian surface is warped with radius that encloses Minkowski space. Using (25) in (16), we readily get from (26) that depends on . In particular at

 Neff(r=rl) = V5α0α′254κ210T3+O(gsM2,g2sM2Nf) V5 ≡ ∫dψdϕ1dϕ2dθ1dθ2sinθ1sinθ2 (27)

Using the exact same argument, we get as at [44]. This implies that there are no effective D3 branes, they have cascaded away at the IR and we are left with only fractional D3 branes.

Note that the holomorphic D7 brane embedding of Ouyang [45] only gives flavor symmetry group. However by considering number of and branes, the chiral symmetry can be realized [68, 58] but then the warp factor (25) will also be modified. In addition, the warp factor (25) only makes sense in the large region which means is large. Thus the fundamental matter fields resulting from the holomorphic embedding have large mass and we do not expect chiral symmetry to hold for massive flavors. As supersymmetry is not broken at zero temperature for the Ouyang embedding, the warp factor (25) corresponds to a gauge theory with massive fundamental matter that is supersymmetric at zero temperature. Recently in [69], the dual to non-supersymmetric confining gauge theory was proposed and the metric (12) is consistent with this proposal with the understanding that in (2.2) be replaced with the perturbed metric which is also Ricci flat. We will analyze possible brane configurations that may give rise to SUSY and non-SUSY gauge theories with fundamental representation matter in section 4. .

The solution (25) leads to divergence of for very large i.e very large energies. But this is expected as the dual theory is a cascading gauge theory with effective color diverging in the UV [44, 59, 70]. However our goal is to study thermal QCD which becomes conformal and free in asymptotically large energies. Thus the dual geometry with warp factor (25) cannot be relevant for a QCD like theory and must be modified for . In [54, 58] a proposal was made to modify the UV dynamics of the cascading gauge theory by introducing anti five branes separated from D5 and D3 branes along with embedding branes to account for fundamental matter and chiral symmetry breaking. The additional branes modify and while the form of the can be obtained from F-theory [66, 67]. Note that the , system behaves as dipoles and thus away from branes, their contribution to flux and axio-dilaton is not divergent and behaves as . One can consider brane embeddings such that the fluxes and axio-dilaton fields in the dual geometry is well defined everywhere in the bulk geometry while the form of the metric (12) remains unchanged. However, the internal metric does not remain Ricci flat anymore and the metric (24) gets corrections of .

The fluxes proposed in [54] essentially give for 6 while the axio dilaton field behaves as for large . Thus for asymptotically large distances, we end up with constant axio-dilaton field and a vanishing NS-NS two form and the geometry behaves as an . This means that addition of anti five branes have slowed down the cascade and the theory has no Landau poles and the associated UV divergences. The warp factor becomes

 h(r,Θ)=1r4∑i(ai(Θ)ri) (28)

for large where . Using (26) one readily gets that and hence does not diverge. As reaches a fixed value with the geometry becoming , the gauge theory reaches a conformal fixed point. However, for supergravity approximation to be valid, we need which means the ’tHooft coupling is still very large . Although we have a conformal field theory at large scale , the gauge group has large number of effective colors with large ’tHooft coupling and thus the gauge theory is not asymptotically free. However, the Yang Mills coupling of our gauge theory behaves the following way

 1g21+1g22 = e−ϕ 1g21−1g22 = e−ϕ∫S2B2 (29)

At asymptotically large energies, that is , and we have with

 limΛ→∞2g2YM=1gs (30)

Now we can take the limit by keeping fixed, which means at asymptotically large energies, just like in QCD. However, since is held fixed at large values, we have large ’tHooft coupling while for QCD, ’tHooft coupling goes to zero.

In summary, the extremal geometry takes the simple form given by (12) while the warp factor is given by (19,25) and (28) for various range of the radial coordinate . The fluxes for various regions of the geometry can be found in [54].

Now to obtain thermodynamic state functions such as free energy, entropy and pressure of the gauge theory that is dual to the geometry , we must obtain the on shell supergravity action on the manifold . However note that, since there is no black hole horizon in the geometry , the Euclidean renormalized on shell action is independent of horizon with independent of . Then using (2) one obtains that the free energy is independent of temperature and (2) readily gives

 s=−∂F∂T=0 (31)

The above result for entropy is consistent with the confined phase of large gauge theory. For confined phase, symmetry is preserved and physical quantities do not depend on the temporal radius at , due to large volume independence [71],[72]. This means free energy is independent of temperature and entropy is zero at . Thus the gauge theory dual to the geometry is in the confined phase. We will later see that for , entropy is nonzero- indicating we have a deconfined phase and degrees of freedom have been released at high temperature.

### 2.2 Non extremal geometry

The non-extremal geometry is a regular black hole with in (2). The internal manifold is typically a resolved-deformed conifold that mimics the symmetries (or breaking of symmetries) of the dual gauge theory. For simplicity, we will consider constant axio-dilaton field, that is there are no seven branes in the dual gauge theory, i.e . We make the following ansatz for the internal metric

 ~gmndxmdxn = a(r)dr2+k(r)~g(1)pqdxpdxq ~g(1)pqdxpdxq ≡ [19(g5)2+16(4∑i=1(gi)2)]+~g′pqdxpdxq (32)

where are defined in (2.1), and represent angular coordinates. Note that when there are no black holes i.e. horizon and no three form flux i.e. , we have and . On the other hand, as we will see explicitly, fluxes enter the Einstein equations with terms of , where

 N≡α0≃54κ210T3Neff(r=rl)V5α′2 (33)

is proportional effective number of D3 branes at certain energy scale . These scalings with horizon and flux strength indicate that at the lowest order, we have

 ~g′pq = O(rhgsM2/N) a(r) ≡ 1+a1(r)=1+O(rhgsM2/N),  b(r)≡1+b1(r)=1+O(rhgsM2/N)

Note that with the two are squashed and topologically the base of the cone is no longer .

Before solving the equations of motions, we would like to discuss some subtleties:

We are studying the gravity dual of the gauge theory and these gravity solutions are generated by fluxes only. The gauge theory lives on N D3 branes and M D5 branes placed at the tip of a regular conifold, the back reactions of the branes will give us a warped geometry. There are fluxes in these gravity solutions whose integral is non-vanishing, however that does not mean there is a function or brane source in such solutions. Actually can be non-zero without any source if the volume of integrated space is non-vanishing anywhere. This is what happens in case, where is a 5D sphere with constant radius. Particularly in Klebanov-Tseytlin solution, the radius of the warped becomes zero at non-zero and becomes ill defined at very small , leading to singularities. This singularity was resolved by Klebanov and Strassler [44] who pointed out that at low energies, the gauge theory superpotential receives quantum corrections [73] and the corresponding dual geometry is no longer the warped regular cone but it must be warped deformed cone whose is non-vanishing at . Thus even though and there is no function in either flux equations or Einstein equations. In section (4), we will outline the brane configuration that may give rise to the dual gravity solutions that we present here.

The Schwarzschild black hole is a vacuum solution of Einstein equations with the stress energy tensor . The horizon radius is an independent parameter and not determined by the Einstein equations. By considering the weak field limit, one can obtain the geometric mass of the black hole from the horizon radius which is a free parameter. Similarly by placing D3 branes at the tip of regular cone, we can generate AdS Schwarzschild black holes and again the black hole mass is a free parameter independent of the number of D3 branes. In our solution discussed below we will find a charged black hole and we interpret the charge and mass arising from the D branes placed at the tip of the regular cone. We will see that the black hole is not a Schwarzschild black hole and the horizon is no longer independent of the number of D branes. However, the solution can be built from the Schwarzschild solution and thus the vacuum solution plays a crucial role in our analysis.

Now lets look at the equations of motions. We first consider the external components of the Einstein equations in (2). Using the form of (5), the first equation in (2) becomes

 Rμν = −gμν[G3⋅¯G348Im˜τ+e−8A−2B∂mα∂mα4]+κ210(Tlocμν−18gμνTloc) (35)

On the other hand, the Ricci tensor in the Minkowski direction takes the following simple form

 Rμν = −12[∂m(gmn∂ngμν)+gmnΓMnM∂mgμν−gmngν′μ′∂mgμ′μ∂ngν′ν] (36)

where and is the Christoffel symbol. Now using the ansatz (2) for the metric, (36) can be written as

 Rtt = e4(A+B)[˜▽2(A+B)−3˜gmn∂nB∂m(A+B)] Rij = −ηije2(2A+B)[˜▽2A−3˜gmn∂nB∂mA] (37)

where the Laplacian is defined in (18).

The set of equations can be simplified by taking the trace of the first equation in (2) and using (2.2). Doing this we get

 ˜▽2(4A+B)−3˜gmn∂nB∂m(4A+B) = e−2A−2BGmnp¯Gmnp12Im˜τ+e−10A−4B∂mα∂mα (38) +k2102e−2A−2B(Tmm−Tμμ)loc

On the other hand using (35) in (2.2), one gets

 Rtt−Rxx=0 (39)

which in turn would immediately imply

 ˜▽2B−3˜gmn∂mB∂nB=0 (40)

Now let’s look at the Bianchi identity. Using (5) in (4) gives

 ~▽2α−3e−2A−2B∂mB∂mα = e2A−B∗6¯G3⋅G312iIm˜τ+2e−6A−3B∂me4A+B∂mα. (41)

Subtracting it from (38) one gets the following

 ˜▽2(e4A+B−α) = e2A−B24Im˜τ|iG3−∗6G3|2+e−6A−3B|∂(e4A+B−α)|2 (42) +3e−2A−2B∂mB∂m(e4A+B−α)+% local source

Next we consider the internal components of Einstein equations. Ricci tensor for the directions take the following form

 Rmn = ˜Rmn+˜gmn˜▽2(A+B)−3˜gmn˜gλk∂λB∂k(A+B) (43) + 3˜▽m∂nB+∂mB∂nB−8∂mA∂nA−2∂(mA∂n)B

where is the covariant derivative given by

 ˜▽mVc=∂mVc−˜ΓbmcVb (44)

for any vector . Here is the Ricci tensor and is the Christoffel symbol for the metric . Using (38) and (40) we find

 ~Rmn = −gmnG3⋅¯G324Im˜τ+Gmab⋅¯Gabn4Im˜τ+∂m˜τ∂n¯˜τ2∣% Im˜τ∣2 (45) −12e−8A−2B∂mα∂nα+8∂mA∂nA −3˜▽m∂nB−∂mB∂nB+2∂(mA∂n)B

To solve for these equations we must first find the three-form flux. For simplicity we will ignore the running of the field, that is . We further take and . With constant and considering fluxes on the deformed cone,